(b) In 30 hours , a sample of plutonium - 243 decays to (1)/(64) of its original amount.Determine the half- life of the isotope. [TC3]

Here's how to determine the half-life of Plutonium-243:**Understanding Half-Life**Half-life is the time it takes for half of a radioactive substance to decay. After one half-life, 50% remains. After two half-lives, 25% remains (half of the remaining 50%), and so on.**Setting up the Equation**We can model radioactive decay with the following equation:N(t) = N₀ * (1/2)^(t/T)Where:* N(t) is the amount of the substance remaining after time t* N₀ is the initial amount of the substance* t is the elapsed time* T is the half-life**Solving for the Half-Life**1. **Fractional Amount Remaining:** The problem states that after 30 hours, 1/64 of the original amount remains. So, N(t) = (1/64) * N₀.2. **Substitute into the Equation:** Substitute the given values into the decay equation: (1/64) * N₀ = N₀ * (1/2)^(30/T)3. **Simplify:** Divide both sides by N₀: 1/64 = (1/2)^(30/T)4. **Rewrite with a Common Base:** Notice that 1/64 can be expressed as (1/2)^6: (1/2)^6 = (1/2)^(30/T)5. **Equate the Exponents:** Since the bases are the same, the exponents must be equal: 6 = 30/T6. **Solve for T:** Multiply both sides by T and then divide by 6: T = 30/6 T = 5 hours**Answer:** The half-life of Plutonium-243 is 5 hours.